Renewable energy continues to receive interest because of the growing energy needs and the limitations associated with classical energy sources in terms of environmental effects and the available reserves. Wave energy is a renewable source that has great potential yet to be utilized. One of the main reasons that wave energy is not yet in a fully commercial state is the lack of economic design of a Wave Energy Converter (WEC). Most of the existing studies in wave energy focus on single Degree-Of-Freedom (DOF) WECs. See U. A. Korde, “On control approaches for efficient primary energy conversion in irregular waves,” OCEANS '98 Conference Proceedings, vol. 3, September 1998, pp. 1427-1431 vol. 3; U. A. Korde et al., “Time domain control of a single mode wave energy device,” Proceedings of the Eleventh International Offshore and Polar Engineering Conference, Stavanger, Norway, 2001, pp. 555-560; F. Fusco and J. Ringwood, IEEE Trans. Sustain. Energy 5(3), 958 (2014); and J. Ringwood et al., IEEE Control Syst. Mag. 34(5), 30 (2014); and J. Scruggs et al., Appl. Ocean Res. 42, 1 (2013). There are a variety of methods for single DOF heave control that range from controls that do not need reactive power, such as the resistive control and latching, to more complex methods that require an actuator with the capability to provide reactive power. See D. G. Wilson et al., “A comparison of wec control strategies,” Sandia National Laboratories, Albuquerque, N. Mex., Tech. Rep. SAND2016-4293, May 2016; A. Babarit et al., Appl. Ocean Res. 26, 227 (2004); and J. Henriques et al., Renew. Energy 45, 31 (2012). Of particular importance among the different control strategies is the well-known complex conjugate control (C3). The C3 criteria are two-fold. First, it resonates the system natural frequencies with the frequencies of the wave excitation force. Second, it adds damping that is equal in magnitude to the system's damping at each frequency. These C3 criteria are shown to guarantee optimal energy harvesting in heave motion. See J. Falnes, Ocean Waves and Oscillating Systems—Linear Interactions Including Wave-Energy Extraction, Cambridge University Press, 2002, ch. 4. The implementation of the C3 criteria, however, has its own challenges; of them is the need to know the frequencies of the wave excitation force. This motivates the wave-by-wave control approach, in which the up-wave elevation measurements are needed. See U. Korde et al., “Approaching maximum power conversion with exergy-based adaptive wave-by-wave control of a wave energy converter,” Proceedings of MTS/IEEE OCEANS, Genova, Italy, May 18-21 2015; and U. A. Korde, “Up-wave surface elevation for smooth hydrodynamic control of wave energy conversion in irregular waves,” Oceans—San Diego, 2013, September 2013, pp. 1-10. Other studies have investigated the use of relative motion of multiple bodies in a heave mode. See J. Ringwood et al., IEEE Control Syst. Mag. 34(5), 30 (2014); and J. Falnes, J. Offshore Mech. Arct. Eng. 121(1), 32 (1999). Davis et al., for instance, presented a sensitivity analysis for a three-body heave WEC to examine how the characteristics of the heave plate and the component masses affect the performance of the system, and concluded that the system is dominated by inertia more than drag. See A. F. Davis et al., “Modeling and analysis of a multi degree of freedom point absorber wave energy converter,” Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, OMAE2014, no. OMAE2014-23475. San Francisco, Calif., USA: ASME, Jun. 8-13 2014.
The main reason for the focus on single DOF WECs is the complexity of designing multiple DOF actuators in addition to the complexity of the control logic itself. On the other hand, other references motivate the use of a multiple-DOF WEC as opposed to a single-mode WEC. See D. V. Evans, J. Fluid Mech. 77, 1 (1976). Evans extended the results of two-dimensional WECs to bodies in channels that accounts for the effect of body orientation on the energy harvesting. See D. Evans, “Some theoretical aspects of three-dimensional wave-energy absorbers,” Proceedings of the first symposium on wave energy utilization, Chalmers University of Technology, Gothenburg, Sweden, 1979, pp. 77-106. French and Bracewell point out that the power that can be extracted from a mode that is antisymmetric to the wave (such as pitch and surge) is twice as much as can be extracted from a mode that is symmetric (such as heave). See M. J. French and R. H. Bracewell, “P.s. frog a point-absorber wave energy converter working in a pitch/surge mode,” Proceedings of The Fifth International Conference on Energy Options: the role of alternatives in the world energy scene, University of Reading, Reading, Berkshire, UK, IEE, 1987. French and Bracewell also point out that roll, yaw, and sway modes are not coupled to the wave and hence conclude that the pitch and surge motions are the most attractive power conversion modes. Moreover, the reactive power in the pitch and surge modes is less than that in the heave mode for the frog system proposed by French and Bracewell. Yavuz recently studied the pitch-surge power conversion. See H. Yavuz, Int. J. Green Energy 8(5), 555 (2011). Yavuz models the pitch-surge motions assuming no heave motion; hence there is no effect from the heave motion on the pitch-surge power conversion. The mathematical model used in Yavuz for the motions in these two DOFs is coupled through mass and damping only; there is no coupling in the stiffness. A Proportional-Derivative (PD) control is used in Yavuz; the controller gains are tuned based on a fourth order polynomial in frequency.
However, a need remains for a WEC that can efficiently extract wave energy over a full range of wave frequencies in the heave, pitch, and surge modes and that does not require prediction of the wave excitation force.